Periodic geodesics and geometry of compact Lorentzian manifolds with a Killing vector field

We study the geometry and the periodic geodesics of a compact Lorentzian manifold that has a Killing vector field which is timelike somewhere. Using a compactness argument for subgroups of the isometry group, we prove the existence of one timelike non self-intersecting periodic geodesic. If the Killing vector field is nowhere vanishing, then there are at least two distinct periodic geodesics; as a special case, compact stationary manifolds have at least two periodic timelike geodesics. We also discuss some properties of the topology of such manifolds. In particular, we show that a compact manifold M admits a Lorentzian metric with a nowhere vanishing Killing vector field which is timelike somewhere if and only if M admits a smooth circle action without fixed points.

Maslov index in semi-Riemannian submersions

We study focal points and Maslov index of a horizontal geodesic gamma : I -> M in the total space of a semi-Riemannian submersion pi : M -> B by determining an explicit relation with the corresponding objects along the projected geodesic pi omicron gamma : I -> B in the base space. We use this result to calculate the focal Maslov index of a (spacelike) geodesic in a stationary spacetime which is orthogonal to a timelike Killing vector field.

Pseudo Focal Points Along Lorentzian Geodesics and Morse Index

Given a Lorentzian manifold (M,g), a geodesic gamma in M and a timelike Jacobi field Y along gamma, we introduce a special class of instants along gamma that we call Y-pseudo conjugate (or focal relatively to some initial orthogonal submanifold). We prove that the Y-pseudo conjugate instants form a finite set, and their number equals the Morse index of (a suitable restriction of) the index form. This gives a Riemannian-like Morse index theorem. As special cases of the theory, we will consider geodesics in stationary and static Lorentzian manifolds, where the Jacobi field Y is obtained as the restriction of a globally defined timelike Killing vector field.

COMPARISON RESULTS FOR CONJUGATE AND FOCAL POINTS IN SEMI-RIEMANNIAN GEOMETRY VIA MASLOV INDEX

We prove an estimate on the difference of Maslov indices relative to the choice of two distinct reference Lagrangians of a continuous path in the Lagrangian Grassmannian of a symplectic space. We discuss some applications to the study of conjugate and focal points along a geodesic in a semi-Riemannian manifold.

Kinetic resolution of a drug precursor by Burkholderia cepacia lipase immobilized by different methodologies on superparamagnetic nanoparticles

Burkholderia cepacia lipase was immobilized on superparamagnetic nanoparticles using three different methodologies (adsorption, chemisorption with carboxibenzaldehyde and chemisorption with glutaraldehyde) and employed in the kinetic resolution of a chiral drug precursor, (RS)-2-bromo-1-(phenyl)ethanol, via enantioselective acetylation reaction. An excellent improvement of lipase catalytical performance was observed. Free B. cepacia lipase gave the ester (S)-2 with poor E-value <30, and after its immobilization to magnetic nanoparticles the E-value was up to >200. The effect of several reaction parameters in the kinetic resolution was studied. The best results for kinetic resolution were obtained using vinyl acetate as acetyl donor and toluene as solvent, typically yielding the ester in high enantiomeric excess (>99%) and E-value (E > 200). Of the three tested immobilization methods, chemisorption with glutaraldehyde was the best one in terms of temperature stability and yield product. (C) 2010 Elsevier B.V. All rights reserved.

Enantioselective transesterification catalysis by Candida antarctica lipase immobilized on superparamagnetic nanoparticles

Lipase B from Candida antarctica can be directly immobilized onto functionalized superparamagnetic nanoparticles, preserving its enzymatic activity in the enantioselective transesterification of secondary alcohols, with excellent results in terms of enantiomeric discrimination. The immobilized enzyme can be easily recovered with a magnet, allowing its reuse with negligible loss of activity. (C) 2009 Elsevier Ltd. All rights reserved

Catalytic properties of thioredoxin immobilized on superparamagnetic nanoparticles

Thioredoxin (Trx1), a very important protein for regulating intracellular redox reactions, was immobilized on iron oxide superparamagnetic nanoparticles previously coated with 3-aminopropyltriethoxysilane (APTS) via covalent coupling using the EDC (1-ethyl-3-{3-dimethylaminopropyl}carbodiimide) method. The system was extensively characterized by atomic force microscopy, vibrational and magnetic techniques. In addition, gold nanoparticles were also employed to probe the exposed groups in the immobilized enzyme based on the SERS (surface enhanced Raman scattering) effect, confirming the accessibility of the cysteines residues at the catalytic site. For the single coated superparamagnetic nanoparticle, by monitoring the enzyme activity with the Ellman reagent, DTNB=5,5`-dithio-bis(2-15 nitrobenzoic acid), an inhibitory effect was observed after the first catalytic cycle. The inhibiting effect disappeared after the application of an additional silicate coating before the AFTS treatment, reflecting a possible influence of unprotected iron-oxide sites in the redox kinetics. In contrast, the doubly coated system exhibited a normal in-vitro kinetic activity, allowing a good enzyme recovery and recyclability. (C) 2011 Elsevier Inc. All rights reserved.

Sample stacking in CZE using dynamic thermal junctions I. Analytes with low dpK(a)/dT crossing a single thermally induced pH junction in a BGE with high dpH/dT

The possibility to compress analyte bands at the beginning of CE runs has many advantages. Analytes at low concentration can be analyzed with high signal-to-noise ratios by using the so-called sample stacking methods. Moreover, sample injections with very narrow initial band widths (small initial standard deviations) are sometimes useful, especially if high resolutions among the bands are required in the shortest run time. In the present work, a method of sample stacking is proposed and demonstrated. It is based on BGEs with high thermal sensitive pHs (high dpH/dT) and analytes with low dpK(a)/dT. High thermal sensitivity means that the working pK(a) of the BGE has a high dpK(a)/dT in modulus. For instance, Tris and Ethanolamine have dpH/dT = -0.028/degrees C and -0.029/degrees C, respectively, whereas carboxylic acids have low dpK(a)/dT values, i.e. in the -0.002/degrees C to+0.002/degrees C range. The action of cooling and heating sections along the capillary during the runs affects also the local viscosity, conductivity, and electric field strength. The effect of these variables on electrophoretic velocity and band compression is theoretically calculated using a simple model. Finally, this stacking method was demonstrated for amino acids derivatized with naphthalene-2,3-dicarboxaldehyde and fluorescamine using a temperature difference of 70 degrees C between two neighbor sections and Tris as separation buffer. In this case, the BGE has a high pH thermal coefficient whereas the carboxylic groups of the analytes have low pK(a) thermal coefficients. The application of these dynamic thermal gradients increased peak height by a factor of two (and decreased the standard deviations of peaks by a factor of two) of aspartic acid and glutamic acid derivatized with naphthalene-2,3-dicarboxaldehyde and serine derivatized with fluorescamine. The effect of thermal compression of bands was not observed when runs were accomplished using phosphate buffer at pH 7 (negative control). Phosphate has a low dpH/dT in this pH range, similar to the dK(a)/dT of analytes. It is shown that vertical bar dK(a)/dT-dpH/dT vertical bar >> 0 is one determinant factor to have significant stacking produced by dynamic thermal junctions.